Finance from the NOVA School of Business and Economics. EXPLOITING THE COINTEGRATION BETWEEN VIX AND CDS IN A CREDIT MARKET TIMING MODEL

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1 A Work Project, presented as part of the requrements for the Award of a Masters Degree n Fnance from the NOVA School of Busness and Economcs. EXPLOITING THE COINTEGRATION BETWEEN VIX AND CDS IN A CREDIT MARKET TIMING MODEL ANDREA RICCIARDI N 976 A Project carred out on the Fnance course, under the supervson of: Paulo Rodrgues NOVA School of Busness and Economcs Paul Beekhuzen & Patrck Houwelng Robeco Quanttatve Strateges January

2 EXPLOITING THE COINTEGRATION BETWEEN VIX AND CDS IN A CREDIT MARKET TIMING MODEL Abstract We nvestgate the contegraton between VIX and CDS ndces, and the possblty of explotng t n an exstng credt market tmng nvestment model. We fnd contegraton over most of the sample perod and the leadershp of VIX over the CDS n the prce dscovery process. We present two methods for ncludng contegraton nto the model. Both strateges mprove the n-sample and out-of-sample model performances, even though out-of-sample results are weaker. We fnd that n-sample better performances are explaned by a stronger contegraton, concludng that n the presence of contegraton our strateges can be proftable n an nvestment model that consders transacton costs. Keywords: Contegraton, VIX, Credt Default Swaps, Pars Tradng. 1. Introducton Ths thess s the result of a sx months nternshp at the Quanttatve Research department of Robeco Asset Management. I focused on the already exstng Credt Default Swaps (CDS) Indces 1 market tmng model, wth the specfc task of mprovng t. The model s called Beta and t s the performance drver of the Robeco Quant Hgh Yeld Fund, n whch around 170 mln are currently nvested. In ths work, we study the contegraton between VIX and CDS ndces, amng at usng t as a sgnal through a varable n the model. The VIX s the mpled volatlty ndex extracted from dfferent optons on the S&P500, and t s often refer to as the fear ndex. It s reasonable to 1 CDS ndces are baskets of sngle-name CDSs. See (Markt, 2014) for more detals. 2

3 beleve n a strong relatonshp between the VIX, measure of the global market rsk, and the CDS ndces, measure of the global credt rsk. We fnd contegraton between VIX and CDS ndces, and we construct two contegraton varables whch, when added to the Beta model, mprove the model performance over the full-sample perod. 1.1.The current Beta model The model conssts of a number of themes amng at forecastng the spread drecton 2. Such themes are grouped nto: Equty, whch gathers nformaton from the equty market and the VIX; Short-Term Trend, ndcaton of the short-term spread momentum; Long-Term Trend, used to capture busness cycle varatons; Seasonal, that apples the Sell n May and Go Away strategy. The outcome of each theme s translated nto a score wth mean 0 and varance 1 (the z-scores), and such scores are combned nto a model score. In order to avod exaggerated contrbuton from any varable, all the scores are capped at ±1. A long poston s taken when the model score s postve, and a short poston s taken when the model score s negatve. The nvestment strategy s weekly Data The Beta model unverses are four: CDX Investment Grade (USIG), CDX Hgh Yeld (USHY), Traxx Man (EUIG), Traxx Crossover (EUHY) 3. The fund nvests n 5-years maturty CDS, the most lqud contracts, and currently just n the HY markets. However, we are nterested n a proftable strategy for both IG and HY, manly for two reasons: frst, the portfolo manager shares the same vson for the determnants movng the IG and HY markets, makng the IG unverse an mportant robustness check; second, n the future the Beta model mght be extended to the IG unverse. 2 See (Houwelng, Beekhuzen, Kyosev, & Van Zundert, 2014) for more detals on the Beta model. 3 We use roll-adjusted versons of the spreads whch take nto account the changes n spread stemmng from the semestral rollng of the ndces. 3

4 The sample startng date s 23-Jun-2004, date where all the ndces returns are avalable. Throughout ths work, we splt the sample nto two sub-perods: an n-sample perod [23-Jun- 2004, 23-Jun-2009], over whch the model s calbrated, and an out-of-sample perod [24-Jun- 2009, 13-Jul-2015], over whch the model s eventually tested. The motvaton behnd ths sample dvson wll be clear later on. Clearly, we need some performance measure to evaluate the strategy. Snce ths work ams at havng a drect mpact on the Beta model, transacton costs must be taken nto account. 4 Our man performance measures wll be the annualzed Net Performance and the annualzed Net IR, defned as the Net Performance over ts volatlty, and both of these measures wll be presented for IG and HY markets separately. We also look at the Turnover, defned as T = 52 mean (mean( Sgnal t PostonSze t Sgnal t 1 PostonSze t 1 )), thus the annualzed mean over markets and tme of the postons dfference n the market. It s also useful to look at the sngle varable performances. Recallng that each varable s translated nto scores between -1 and 1, we defne the z-performance as the annualzed average of the scores tmes return. Such measure can gve an ntal nsght on the contrbuton of the sngle varable. Our sngle varables evaluaton performances wll be z-perfomances, z- volatlty and z-ir, for IG and HY. The rest of the thess s organzed as follows. Secton 2 gves an overvew of the exstng lterature. Secton 3 descrbes the methodology that wll be used throughout ths work. In Secton 4 we report n-sample and out-of-sample results, eventually dscussng the dfferences. Secton 5 presents some deas for future researches and Secton 6 concludes. Numbered graphs, tables and an addtonal dervaton are reported n Appendx A. Other graphs and tables are reported n Appendx B. 4 Robeco has ts own transacton costs model whose detals go beyond the purpose of ths work. 4

5 2. Lterature Revew The relatonshp between credt rsk and equty volatlty has been wdely studed n the lterature, startng from (Merton, 1974) structural credt model. When studyng ndces nstead of sngle credt spreads, the asset volatlty s replaced by VIX, recognzed measure for the market fear, as n (Colln-Dufresne & Goldsten, 2001) and (Shaefer & Strebulaev, 2008). However, even though the close relatonshp between VIX and CDS s wdely known, ther contegraton nvestgaton s almost not present n the lterature. As far as we know, (Fguerola-Ferrett & Paraskevopoulos, 2013) s the only paper studyng the contegraton between VIX and CDS. They fnd contegraton between VIX and Traxx markets, and they propose a pars tradng strategy n the VIX futures and 5 years Traxx that brngs to abnormal postve returns. Ths thess adds value to (Fguerola-Ferrett & Paraskevopoulos, 2013) n several ways. Frst, n the out-of-sample robustness check, they smply recalculate the contegraton relatonshp usng future nformaton they dd not have at that pont n tme. Our out-of-sample s meant as a pure out-of-sample test: we perform weekly rollng regressons usng just nformaton we had at that pont n tme, thus calculatng every week the contegraton relathonshp. Second, even though (Fguerola-Ferrett & Paraskevopoulos, 2013) acknowledge the Gonzalo-Granger measure as a measure of VIX leadershp n the prce dscovery process, they do not take advantage of t n the tradng strategy. Instead, our contegraton varables wll nclude such nformaton, crucal to understand the contrbuton of each seres to the equlbrum readjustment. Thrd, we extend the data sample from to In the lterature, contegraton pars are usually exploted wth pars tradng strateges. Snce our model can nvest just n one of the two assets, the CDS, we develop a unlateral pars tradng strategy, method present n the hedge fund lterature thanks to (Altucher, 2004), but 5

6 not n the academc s one. We follow the nvestment model presented by (Caldera & Moura, 2012), who apply a pars tradng strategy on the contegraton pars dentfed n the Brazlan stock market. Unlke most of the lterature, where nvestors explot contegraton wthn a hgh-frequency framework, such as (Mao, 2014) and (Hanson & Hall, 2012), we wll explot t wth a long term varable. 3. Methodology In ths secton, we llustrate the methodology for our work. We start by testng for statonarty, snce non-statonarty s an essental requrement for contegraton. Then we perform the contegraton Johansen test and we ntroduce the Gonzalo-Granger measure, ndcator of the VIX leadershp over the CDS n the prce dscovery process. Fnally, we construct two contegraton varables: a dscrete one, and a contnuous one Non-statonarty Before testng for contegraton, we must assure that we are dealng wth non-statonary tme seres. Practtoners often consder VIX as mean-revertng. However, VIX s smply the mpled volatlty extracted from the nearby S&P 500 ndex optons, usng a wde range of strkes. Thus, the statstcal propertes of VIX stem from the dstrbuton of weghted average opton prces. (Fguerola-Ferrett & Paraskevopoulos, 2013) emprcally show that the VIX s not mean-revertng. We test ths hypothess by performng the Augmented Dckey-Fuller (ADF) test wth drft for unt roots and we fnd statonarty over the full-sample. However, for our contegraton analyss we wll use a 5-years rollng wndow: by usng such wndow, the VIX turns out to be non-statonary Contegraton After testng for unt roots, we test for contegraton between CDS ndces spreads and VIX approachng the Johansen test. Let s t denote the CDS spread at tme t for market, and let v t 6

7 be the VIX spot at tme t. We wonder f there exsts a non-trval vector [γ 0, γ 1 ] such that the process {s t γ 0 γ 1 v t } t s statonary. The vector [1, γ 0, γ 1 ] s called the contegraton vector and the process {z t } defned as z t s t γ 0 γ 1 v t (1) s called the contegraton relatonshp. If such vector exsts, the spread can be replcated by borrowng/nvestng γ 0 n the rsk-free asset and by buyng/sellng γ 1 unts of the asset v VECM and Gonzalo-Granger measure Contegraton s a statstcal property that s wdely exploted n pars tradng strateges. In such a framework, when process (1) wdens, we can short the wnner and long the loser, confdent n a reverson to the long-run equlbrum. Snce our model does not nvest n one of the two assets the VIX, our strategy wll follow the pars tradng strategy but unlaterally, meanng that we open postons just on the spread. A legtmate objecton to ths knd of strategy reles on the fact that pars tradng works because a pars of assets s ndeed traded. Contegraton suggests that the gap between the two assets wll eventually go back to ts equlbrum, but t gves no ndcaton about whch asset wll contrbute the most to the resettlement of such relatonshp. Therefore, by nvestng n just one asset, one could argue that a pror we lose half of the strategy s power. (Fguerola- Ferrett & Paraskevopoulos, 2013) show that the CDSs do all the work n terms of equlbrum readjustment. They explan such predomnance wth a hgher number of partcpants (thus, hgher lqudty) n the VIX futures market. We follow the model developed by (Fguerola-Ferrett & Paraskevopoulos, 2013) descrbng the nteracton between trades n the CDS and VIX market, and we test for VIX predomnance n our data sample. Such model leads to a Vector Error Correcton Model (VECM) framework such as: 7

8 ( Δs t Δv ) = (α 1 t α ) z t ( u s, t v, u ) (2) t where α 1 and α 2 are known as adjustment coeffcents and u t s, and u t v, are the error terms. If the two coeffcents are both statstcally sgnfcant, they must have opposte sgn, snce a devaton from equlbrum wll be readjusted wth opposte movements of spread and VIX. Moreover, a α 2 not sgnfcantly dfferent from zero ndcates that VIX does not adjust to the spread, meanng that VIX domnates the CDS spreads n the prce dscovery process. Such result would justfy our unlateral pars tradng strategy. Followng (Blanco, Brennan, & Marsh, 2005), we can ntroduce a measure of VIX leadershp n the prce dscovery process, nspred by (Gonzalo & Granger, 1995): GG VIX = α 1 α 2 α 1 (3) The Gonzalo-Granger measure s useful when both coeffcents are statstcally sgnfcant and have opposte sgn. In such a case, GG VIX [0,1], and a value close to 1 ndcates the VIX leadershp, whereas a value close to 0 ndcates that the spread domnates the VIX Tradng strategy mplementaton In ths secton, we dg nto the constructon of our two contegraton varables. We choose a wndow of D days over whch the contegraton relatonshp s calbrated. Per each day, we compute the parameters γ 0 and γ 1 n (1) usng the data of the past D days, endng up havng tme-dependent parameters γ,d 0 (t) and γ,d 1 (t) and as a consequence a tmedependent contegraton relatonshp z,d (t). Parameters are estmated wth the Johansen method, and lags are chosen followng the AIC crteron. Acknowledgng the regresson estmates senstvty to outlers, we set an nterval wthn whch the parameters should reasonably lay. If the new estmates overstep the nterval, we look backwards usng the 8

9 closest parameters satsfyng the constrants. In order to make z,d (t) comparable across varables, we dvde t by the movng standard devaton σ,d z (t). Notce that we do not have to subtract t by the mean, snce by constructon process (1) s statonary wth constant mean 0. Recallng that the nvestment strategy s weekly, n order to avod any possble day-of-theweek effect we take the average of z,d (t) over the last 5 (workng) days, endng up wth a statonary process z,d (t). We then gve two alternatve ways of buldng the Contegraton VIX (CV) varable. The frst method tackles the problem n a way smlar to the pars tradng lterature, as for nstance (Gatev, Goetzmann, & Rouwenhorst, 2006) and we call t CV dscrete. Wth the second method we buld a more contnuous varable, smlarly to most of the varables at Robeco Quanttatve Strateges, and we call t CV contnuous. The two varables are descrbed n the next sectons. In any case, gven the nature of the varable, we add t to the Equty basket. In the Equty basket there s already a varable, called VIX Trend, whch looks at the trend nformaton from the VIX. Thus, we checked for correlaton between such varable and our contegraton varables. The varables show low-mutual correlaton, as reported n Appendx B Dscrete varable When spread and VIX depart too much from each other, we bet the spread wll move towards the replcaton strategy γ,d 0 (t) + γ,d 1 (t)v(t). The devaton from each other s measured by z,d (t). Recallng that z s statonary wth mean 0 and unt varance, the mplementaton rule s the followng (see Fgure 1 for a graphcal descrpton): Let k > 0. Go long whenever z,d > k and offset the poston when z,d < 0; go short whenever z,d < k and offset the poston when z,d > 0. Moreover, offset the (4) poston f contegraton has not been found n the past D days. 9

10 Let V be the varable descrbed by (4). So far we defned the drecton of V, but not the poston sze. We set the varable to zero when the sgnal s neutral, and to ± M when we open a long or short poston. M must be chosen n such a way that the varable weght (defned as the average of the scores absolute values) n the basket s the same as the other two varables (the Equty Trend and the VIX Trend). We calbrated the value wth Monte Carlo smulatons, resultng n M = 2.4. See Appendx A for the dervaton. In (4), k s a parameter ndcatng how far we have to be from equlbrum before swtchng on the sgnal. After n-sample calbratons, we chose k = 1.5. Notce that when we do not fnd contegraton n the past D days we set the varable to zero. Ths s motvated manly by two reasons. Frst, f we dd not fnd contegraton ths means that we dd not fnd any par (γ 0, γ 1 ) such that process (1) s statonary, resultng n based (spurous) coeffcents estmates from equaton (2) and based Gonzalo-Granger measures. Second, ths s the method generally used n the pars tradng lterature. The varable wll be composed of two factors:,d CV dscrete (t) = V,D (t) GG,D (t), (5) where GG,D (t) s the Gonzalo-Granger measure at tme t for market defned by (3), and V,D (t) s the varable descrbed by rule (4). We decded to nclude the Gonzalo-Granger measure n the varable snce t s crucal nformaton for our unlateral pars tradng strategy. The bgger the measure, the more chances we have that the spread wll follow the VIX, nstead of the other way around. Fnally, the varable s capped at ± Contnuous varable The varable constructed above s dscrete, n the sense that t assumes ether 0 or M GG t. We can also defne a contnuous varable usng bascally the process z,d (t). However, all the 10

11 varables n the model are capped at 1. Ths means that, assumng k > 1, the varable at k and 1 would have the same score 1, gven as a consequence the same weght to z,d = 1 and to z,d = k, crtcal value n the prevous varable. Snce we are tryng to explot the same effect from both varables, for consstency we dvde the process by k, such that the varable wll have score 1 just from k onwards. Lastly, we multply the varable by the Gonzalo- Granger measure:,d z,d (t) CV contnuous (t) = mn (max ( GG,D (t), 1), 1) (6) k An nsghtful scheme of the constructon of the two varables s reported n Appendx B. 4. Results In ths secton, we present the man n-sample and out-of-sample results. We fnd stronger performances n the n-sample perod wth respect the out-of-sample, and we eventually gve an explanaton for such better performances In-sample Non-Statonarty Confrmng (Fguerola-Ferrett & Paraskevopoulos, 2013) results, we fnd VIX and spreads to be non-statonary wthn the n-sample perod. However, f we take the full-sample perod, the ADF test rejects the null hypothess of a unt root for the VIX seres. In order to better understand what s gong on, we plot the ADF statstcs over tme n Fgure 2. At each pont n tme the ADF statstc s calculated lookng backwards from 1998 untl the current tme. Fgure 2, whle showng the non-statonarty of the spreads, confrms our suspcons: for most of the sample, the null hypothess of unt root n the VIX s rejected. Recallng that nonstatonarty s a necessary property for contegraton, ths thess should end here. 11

12 However, our varable s not constructed by consderng the seres from the begnnng of the sample perod, nstead t looks backwards just D days. Therefore we are not nterested n nonstatonarty over the full sample, but n non-statonarty n a movng wndow of D days. Thus, we need to calculate the ADF statstcs by lookng backwards D days nstead of lookng from the begnnng of the sample. We choose 5 years as a movng wndow 5 and we plot the ADF statstcs over tme n Fgure 3. We can see how thngs get sgnfcantly better. Even though there are stll some perods over whch the seres seems statonary, the ADF test generally does not reject the null of a unt root, openng the door for contegraton. We can conclude that the VIX dsplays persstence change over tme Contegraton A prelmnary contegraton analyss can be made by performng the contegraton tests over the n-sample perod. Table 1 and Table 2 report the test results and the coeffcent estmates. Both tests show strong contegraton between VIX and CDSs. Table 1 shows that wth the Johansen estmates we replcate the spreads by borrowng the rsk free asset and nvestng n γ 1 unts of VIX. Table 2 reports the adjustment coeffcents for each market and the respectve Gonzalo-Granger measure, defned by (3). As underlned n Secton 3.3, a Gonzalo-Granger measure close to one ndcates a strong leadershp of VIX n the prce dscovery process. For the HY markets α 2 s not even sgnfcant, sayng that the CDSs do all the adjustment towards the equlbrum, and ths s a very promsng sgn for our strategy. For the USIG market the results are slghtly less strong, but they stll show a strong predomnance of VIX over the CDSs n the prce dscovery. By lookng at Fgure 5, we can have a vsual dea of how the dscrete varable works. 6 Fgure 5 refers to the Johansen parameters estmates reported n Table 1. Clearly, such parameters 5 The wndow has been calbrated n-sample by lookng at a wde range from 80 days to 5 years. 6 For the other markets, see Appendx B. 12

13 are estmated by lookng forward at the whole n-sample perod, therefore, at each pont n tme, Fgure 5 uses nformaton we dd not have at that moment. The yellow strpes ndcate a bearsh sgnal from the varable (that s, bettng that the spread ncreases), the blue strpes a bullsh sgnal, and the whte strpes a neutral sgnal. The only dfference between the dscrete and the contnuous varables occurs when the dscrete varable gves a neutral sgnal. In such crcumstances, the contnuous varable gves a sgnal based on whether the spread s above or below the replcaton strategy. However, the dstance between these two processes s not enough to swtch on the dscrete varable. See Appendx B for the contnuous varable graphs Backtest Let s now move to the performances secton. Even though a good performance s not a suffcent condton for statng a varable as a good one for our model (n fact we need also economcal meanng, robustness, etc.), t s defntely a necessary one. We start wth the n-sample backtest, whch of course could not have been a doable nvestment strategy at that tme, snce for t we use nformaton we stll dd not have. As mentoned n Secton 1.2, we start by studyng the mpact of the sngle varable. Results are shown n Table 3. Here, performances are meant as z-performances. See Secton 1.2 for the defnton. Consderng that the other varables n the Beta model have a z-ir around 0.5 (see Appendx B), both varables look very promsng. From the model pont of vew, our benchmark wll be the current Beta model. The most mportant statstcs wll be Net Performance and Net IR, but also Turnover wll be carefully watched (see Secton 1.2 for ther defnton). Table 4 reports the n-sample model performances. The contegraton varables mprove sgnfcantly the performances. A performance of 0.31 ndcates an annual average return of 31 bass ponts. Even though the dscrete varable s clearly the best, the contnuous varable stll brngs a bg mprovement to the Beta model, especally for the IG markets. Fgure 4 shows the n-sample cumulatve 13

14 sgnal performances of the Beta model and the model wth the dscrete contegraton varable. We can notce that the varable s greatest contrbuton happens to be at the begnnng of the subprme crss, meanng that the VIX spotted the crss before the CDSs dd Out-of-sample The out-of-sample backtest s a fundamental test to prevent data mnng and to assess the qualty and robustness of the varables, because here we are usng just nformaton we had at that pont n tme, makng of ths strategy an nvestable strategy that we could have used from 2009 to Recall that every week we are lookng 5 years backwards, calculate the contegraton parameters, and add the contegraton varable to the Equty basket. Agan, we start by lookng at the sngle varables performances, reported n Table 5. Comparng the above results wth Table 3, the varables seem clearly not as strong as n the n-sample perod, therefore we do not expect an outstandng performance at a model level. However, the varable s contrbuton s stll overall postve. It s nterestng to look at how often the varable s set to zero due to non-contegraton. By takng the average over the markets, we see that ths happens 731 tmes n the out-of-sample perod, resultng n a fracton of 11% of the total observatons, but 717 out of 731 tmes the non-contegraton comes from the EUIG market. Thus, the GG measure for the EUIG market turns out to be the most volatle and less relable one, as we can see from the plot of the GG measure n Appendx B. Table 6 shows the models performances. Results are not as strong as n-sample. However, whereas results for HY markets do not bascally change, IG performance and IG IR double, meanng that the varable stll adds value to the model Dfference between n-sample and out-of-sample Even though the out-of-sample performances are good, they are far from the outstandng nsample results. We test three possble reasons for ths: 14

15 1. VIX s statonary n the out-of-sample perod; 2. contegraton s as strong as n the n-sample perod, but t s not explotable n an nvestment strategy that does not use a forward-lookng wndow; 3. contegraton n the out-of-sample perod s not as strong as n the n-sample perod. In ths secton, we show that the thrd explanaton holds. Thus, our strateges can be proftable wthout knowng the future, and results are mpressve when contegraton s strong, and less mpressve, but stll postve, when contegraton s weak Test for VIX non-statonarty n the out-of-sample perod We frst check whether the VIX non-statonarty stll holds n the out-of-sample perod. By plottng the ADF statstc over tme n Fgure 6, we can see that, even though the test rejects the null hypothess of a unt root n the frst half of 2014, the process seems generally nonstatonary, smlarly to the n-sample results plotted n Fgure 3. Fgure 6 confrms the VIX as a non-statonary seres n the short term. Thus, the dfference n results between n-sample and out-of-sample s not due to a statonarty of the VIX Test for strategy s feasblty wthout a forward-lookng wndow In order to test reason (2), we can perform an a posteror analyss by computng the contegraton parameters for the out-of-sample perod n the same way we computed them for the n-sample. Namely, we perform the Johansen test over the whole out-of-sample perod, endng up wth just one constant par of contegraton parameters per market, γ 0 and γ 1, whch wll be used for constructng the contegraton varables. We then backtest ths model (ftted model) n the out-of-sample perod. Results are shown n Table 7, where are reported the outperformances of the model wth respect the Beta n the n-sample and the out-of-sample perod. The thrd column refers to the outperformance of the ftted model just descrbed. Values n Table 7 are obtaned by 15

16 subtractng the statstcs of the current Beta model from the statstcs of the model wth the CV dscrete varable. By comparng columns 2 and 3 of Table 7, we see that the mprovements comng from the ftted model are of the same out-of-sample model s order. As a consequence, even wth a forward lookng wndow performances would not have changed. Thus, we do not need a forward lookng wndow to explot the contegraton. On the other hand, we are gong to show that a strategy wth weekly rollng regresson would stll have been proftable durng the n-sample perod. In order to test t, we need to reduce our movng wndow. By choosng 3 years, we can estmate the model n the new out-of-sample perod (wthn the orgnal n-sample) from 2007 to Results are shown n Table 8. Improvements get closer to the orgnal n-sample mprovements. IG market performances slghtly worsen, but the HY market performances and IRs ncrease by one thrd, bascally the same mprovement reported n Table 4 for the n-sample back tests. One could argue that Table 8 could be explaned by a better performance of the varable wth 3 years movng wndow. Such argument s refuted by Table 9, where the models wth 3 and 5 years movng wndow are backtested for the full sample perod. The 5-years movng wndow model works better than the 3-years one, although results are smlar. Summarzng, we do not need a forward lookng wndow to explot the contegraton, and we can buld a proftable nvestment strategy explotng contegraton wthout knowng the future. As a result, argument (2) s dscarded Test for contegraton n the out-of-sample perod Fnally, we test for argument (3) by lookng at the contegraton n the out-of-sample perod. Johansen statstcs for the n-sample and out-of-sample perods are reported n Table Whereas the IG markets stay more or less at the same levels (EUIG s weaker contegraton s 7 See Appendx B for a plot of the contegraton statstcs over tme. 16

17 balanced by the USIG s stronger contegraton), the HY markets show much hgher statstcs n the n-sample perod. Ths stronger contegraton results n a better performance for HY n the n-sample perod, as shown by the frst two columns of Table 7. For the IG markets, the outperformance of the contegraton varable s lower n the out-of-sample perod, but not dramatcally lower. For the HY markets, the net performance goes from a n the nsample perod to a poor n the out-of-sample perod. As a consequence, argument (3) holds. Therefore, a tradng strategy explotng the contegraton between VIX and CDS spreads s feasble, and results depend on the strength of the contegraton. 5. Future Research In our model, the bet szes of the contegraton varables do not take nto account how far the assets are from equlbrum. When the msprcng s strong, the poston sze wll always be M for the dscrete varable, and 1 for the contnuous one (tmes the GG measure), resultng n a varable whose nature s statc from the dstance-from-equlbrum pont of vew. However, t mght be nterestng to use dynamc bet szes. (Jurek & Yang, 2006) show the exstence of a crtcal level of msprcng beyond whch an optmal allocaton requres a reducton n the bet sze. When appled to Samese twn shares, such dynamc bet szes result n a sgnfcant mprovement n the Sharpe rato relatve to a smple threshold rule lke ours one. Snce the Beta model nvests just on CDSs, we developed a unlateral pars tradng strategy. Although such choce s justfed by the GG measure (3), that for most of the sample ndcates the VIX leadershp over the CDSs n the prce dscovery process, the results of a pure pars tradng strategy that takes postons n both CDS ndces and VIX futures would be nsghtful, n order to understand the full power of the strategy. 17

18 It mght also be useful to buld a basket of contegraton varables wth dfferent look back horzons. In our model, we chose a 5-years movng wndow, after an n-sample calbraton. However, extractng nformaton from dfferent horzons could add value to the varable, especally n terms of robustness. A further nterestng follow-up for the model would be the contegraton analyss among the spreads over regon and/or credt ratng. Trends nformaton from dfferent markets s already present n the Beta model but, as we have seen throughout ths work, f the seres are contegrated we can obtan addtonal sgnals not captured by the trend varables. 6. Conclusons Ths thess nvestgates the contegraton between VIX and CDS ndces, amng at mprovng the current Robeco CDS market tmng model by addng a Contegraton varable. After testng for non-statonarty (VIX shows persstence change over tme), we fnd contegraton over most of the sample perod ( ). We make use of the VECM (2) to defne the Gonzalo-Granger measure (3). Such measure mostly assumes values close to one (ts average over tme and markets s 0.86, as reported n Appendx A), meanng that VIX leads CDSs n the prce dscovery process. Ths result justfes the use of a unlateral pars tradng strategy. We then construct two contegraton varables amng at explotng the same effect, comparng eventually the results: a dscrete varable and a contnuous varable. We splt the sample n an n-sample perod, over whch we use a forward lookng wndow to calbrate the parameters, and an out-of-sample perod, over whch weekly regressons buld the varable wthout usng any future nformaton. The varables mprove the current model n both the nsample and out-of-sample perods, after transacton costs. However, mprovements are sgnfcantly lower n the out-of-sample perod when compared to the n-sample ones. We prove that such dfference n the performances s explaned by a stronger contegraton durng 18

19 the n-sample perod, and not by mpossblty n mplementng the strategy wthout knowng the future. Concludng, the exstng contegraton between VIX and CDS ndces can be exploted n a proftable tradng strategy after transacton costs, and profts ncrease wth the strength of the contegraton. References Altucher, J. (2004). Trade Lke a Hedge Fund. Wley. Blanco, R., Brennan, S., & Marsh, I. W. (2005). An Emprcal Analyss of the Dynamc Relaton between Investment-Grade Bonds and Credt Default Swaps. The Journal of Fnance, 60: Caldera, J. F., & Moura, G. (2012). Selecton of a Portfolo of Pars Based on Contegraton: The Brazlan Case. Colln-Dufresne, P., & Goldsten, R. S. (2001). Do Credt Spreads Reflect Statonary Leverage Ratos? The Journal of Fnance, 56: Fguerola-Ferrett, I., & Paraskevopoulos, I. (2013). Parng Market Rsk and Credt Rsk. SSRN Workng Paper Seres. Gatev, E., Goetzmann, W. N., & Rouwenhorst, K. G. (2006). Pars Tradng: Performance of a Relatve- Value Arbtrage Rule. Revew of Fnancal Studes, Gonzalo, J., & Granger, C. (1995). Estmaton of Common Long-Memory Components n Contegrated Systems. Journal of Busness & Economc Statstcs, 13: Hanson, T. A., & Hall, J. (2012). Statstcal Arbtrage Tradng Strateges and Hgh Frequency Tradng. SSRN Houwelng, P., Beekhuzen, P., Kyosev, G., & Van Zundert, J. (2014). Beta Model. Jurek, J., & Yang, H. (2006). Dynamc Portfolo Selecton n Arbtrage. EFA 2006 Meetngs Paper. Markt. (2014). Markt credt ndces: A prmer. Retreved from Merton, R. C. (1974). On the Prcng of Corporate Debt: The Rsk Structure of Interest Rates. The Journal of Fnance, 29: Mao, G. (2014). Hgh Frequency and Dynamc Pars Tradng Based on Statstcal Arbtrage Usng a Two-Stage Correlaton and Contegraton Approach. Internatonal Journal of Economcs and Fnance. Shaefer, S. M., & Strebulaev, I. A. (2008). Structural models of credt rsk are useful: Evdence from hedge ratos on corporate bonds. Journal of Fnancal Economcs, 52:

20 APPENDIX A A.1. Fgures Fgure 1: z process underlyng the dscrete varable for the EUHY market. Yellow strpes ndcate a short sgnal, blue strpes a long one. Fgure 2: In-sample spreads and VIX ADF statstcs. The statstcs at tme t refer to the perod [1998,t]. A statstc below the crtcal value ndcates the rejecton of the null hypothess of unt root n favor of the alternatve of statonarty 20

21 Fgure 3: In-sample VIX ADF statstc. The statstcs at tme t refer to the perod [t 5 years, t]. A statstc below the crtcal value ndcates the rejecton of the null hypothess of unt root n favor of the alternatve of statonarty. Fgure 4: In-sample cumulatve sgnal performance comparson between Beta and Beta wth the dscrete contegraton varable. 21

22 Fgure 5: Dscrete varable sgnals for the EUHY market. In gold, γ 0 + γ 1 v represents the spread replcaton stemmng from the contegraton. Fgure 6: Out-of-sample VIX ADF statstc. The statstcs at tme t refer to the perod [t 5 years, t]. A statstc below the crtcal value ndcates the rejecton of the null hypothess of unt root n favor of the alternatve of statonarty. 22

23 A.2. Tables Table 1: Johansen contegraton results and coeffcent estmates. Assets Johansen Test Contegrated USIG, VIX yes USHY,VIX yes EUIG, VIX yes EUHY, VIX yes Table 2: Adjustment coeffcents and Gonzalo-Granger measure. Assets Adjustment Coeffcents USIG, VIX * * 0.61 USHY, VIX * EUIG, VIX * * 0.96 EUHY, VIX * Table 3: In-sample sngle varables z performances. Varable Perf IG Perf HY Vol IG Vol HY IR IG IR HY Contegraton VIX dscrete Contegraton VIX contnuous Table 4: In-sample models performances. Model Net Perf IG Net Perf HY Net IR IG Net IR HY Turnover IG Turnover HY Beta Dscrete Contnuous Table 5: Out-of-sample sngle varables z performances. Varable Perf IG Perf HY Vol IG Vol HY IR IG IR HY Contegraton VIX dscrete Contegraton VIX contnuous Table 6: Out-of-sample model performances. Model Net Perf IG Net Perf HY Net IR IG Net IR HY Turnover IG Turnover HY Beta Dscrete Contnuous

24 Table 7: Beta model outperformance when ncludng dscrete varable. Contegraton VIX dscrete - Beta In-sample Out-of-sample Out-of-sample ftted Net Performance IG Net Performance HY Net IR IG Net IR HY Table 8: Out-of-sample models results for contegraton varables wth a lookback wndow of 3 years nstead of 5. 3 years Movng Wndow, Model Net Perf IG Net Perf HY Net IR IG Net IR HY Turnover IG Turnover HY Beta Contegraton VIX dscrete Contegraton VIX contnuous Table 9: Full-sample model results wth 3 and 5 years movng wndow. Full sample: Model Net Perf IG Net Perf HY Net IR IG Net IR HY Movng Wndow Beta Contegraton VIX dscrete Contegraton VIX contnuous Contegraton VIX dscrete Contegraton VIX contnuous Table 10: Johansen statstcs for the n-sample and out-of-sample perods. Contegraton Johansen Statstc Perod USIG USHY EUIG EUHY In-sample Out-of-sample

25 A.3. CV dscrete score computaton: Monte Carlo smulatons We compute the value M n (4) va Monte Carlo smulatons. The CV varable s added to the Equty basket, where other two varables are already present. Therefore we generate vectors of two multvarate normal varables followng the dstrbuton N 2 (0, Σ), where Σ s the hstorcal covarance matrx of the two varables Equty Trend and VIX Trend. We compute the hstorcal correlaton ρ for the full sample, gettng a value of ρ = After generatng the multvarate vector, we cap each varable at ±1 and we calculate the average of the absolute value of these two varables. By takng the average over the smulatons, we get the expected contrbute of each varable to the Equty basket, that s C = However, ths s not the value of M, snce CV dscrete assumes also value 0. Therefore ts expected contrbuton must be larger than M. Denotng by p the probablty P(CV dscrete = 0), CV expected contrbuton s E CV dscrete = p 0 + (1 p) M E(GG(t)). We can compute the expected Gonzalo-Granger measure by takng ts average over tme and markets. The result s E(GG(t)) = Snce the varable has to gve the same contrbute as the other two, we set 0.86 M(1 p) = C M = C 0.86 (1 p) Thus we have to compute p. We can calbrate t by extractng the mpled probablty from the n-sample perod, by lookng at how many tmes the varable equals 0. The varable s calculated wth the Johansen method wth a wndow of 5 years. Our estmated probablty s p = 0.7, gvng a value of M =